Nonlinear thermodynamical formalism
Annales Henri Lebesgue, Volume 6 (2023), pp. 1429-1477.



We define a nonlinear thermodynamical formalism which translates into dynamical system theory the statistical mechanics of generalized mean-field models, extending the investigation of the quadratic case in one or more potentials by Leplaideur and Watbled.

We prove a variational principle for the nonlinear pressure and we characterize the nonlinear equilibrium measures and relate them to specific classical equilibrium measures.

In this non-linear thermodynamical formalism, as for mean-field theories of statistical mechanics, several kind of phase transitions appear, some of which cannot happen in the linear case. Our techniques can deal with known cases (Curie–Weiss and Potts models) as well as with new examples (metastable phase transition).

Finally, we apply some of these ideas to the classical, linear setting proving that freezing phase transitions can occur over any zero-entropy invariant compact subset of the phase space.


[Bal00] Baladi, Viviane Positive transfer operators and decay of correlations, Advanced Series in Nonlinear Dynamics, 16, World Scientific, 2000 | DOI | Zbl

[BH21] Barreira, Luis; Holanda, Carllos E. Higher-dimensional nonlinear thermodynamic formalism (2021) (

[BK83] Brin, Michael; Katok, Anatole On Local Entropy, Geometric dynamics, Proc. int. Symp., Rio de Janeiro/Brasil 1981 (Lecture Notes in Mathematics), Volume 1007, Springer, 1983, pp. 30-38 | MR | Zbl

[BL13] Bruin, Henk; Leplaideur, Renaud Renormalization, thermodynamic formalism and quasi-crystals in subshifts, Commun. Math. Phys., Volume 321 (2013) no. 1, pp. 209-247 | DOI | MR | Zbl

[BL15] Bruin, Henk; Leplaideur, Renaud Renormalization, freezing phase transitions and Fibonacci quasicrystals, Ann. Sci. Éc. Norm. Supér., Volume 48 (2015) no. 3, pp. 739-763 | DOI | MR | Zbl

[Bow08] Bowen, Rufus Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, 470, Springer, 2008 | DOI | Zbl

[BS01] Barreira, Luis; Saussol, Benoît Variational principles and mixed multifractal spectra, Trans. Am. Math. Soc., Volume 353 (2001) no. 10, pp. 3919-3944 | DOI | MR | Zbl

[BSS02] Barreira, Luis; Saussol, Benoît; Schmeling, Jörg Higher-dimensional multifractal analysis, J. Math. Pures Appl., Volume 81 (2002) no. 1, pp. 67-91 | DOI | MR | Zbl

[Buz97] Buzzi, Jérôme Intrinsic ergodicity of smooth interval maps, Isr. J. Math., Volume 100 (1997), pp. 125-161 | DOI | MR | Zbl

[Cli13] Climenhaga, Vaughn Topological pressure of simultaneous level sets, Nonlinearity, Volume 26 (2013) no. 1, pp. 241-268 | DOI | MR | Zbl

[Cli14] Climenhaga, Vaughn The thermodynamic approach to multifractal analysis, Ergodic Theory Dyn. Syst., Volume 34 (2014) no. 5, pp. 1409-1450 | DOI | MR | Zbl

[Cli18] Climenhaga, Vaughn Specification and towers in shift spaces, Commun. Math. Phys., Volume 364 (2018) no. 2, pp. 441-504 | DOI | MR | Zbl

[CS09] Cyr, Van; Sarig, Omri Spectral gap and transience for Ruelle operators on countable Markov shifts, Commun. Math. Phys., Volume 292 (2009) no. 3, pp. 637-666 | MR | Zbl

[EW90] Ellis, Richard S.; Wang, Kongming Limit theorems for the empirical vector of the Curie–Weiss–Potts model, Stochastic Processes Appl., Volume 35 (1990) no. 1, pp. 59-79 | DOI | MR | Zbl

[FO88] Föllmer, Hans; Orey, Steven Large deviations for the empirical field of a Gibbs measure, Ann. Probab., Volume 16 (1988) no. 3, pp. 961-977 | MR | Zbl

[Fra15] Fraser, Jonathan M. First and second moments for self-similar couplings and Wasserstein distances, Math. Nachr., Volume 288 (2015) no. 17-18, pp. 2028-2041 | DOI | MR | Zbl

[GKLMF18] Giulietti, Paolo; Kloeckner, Benoît R.; Lopes, Artur O.; Marcon Farias, Diego The calculus of thermodynamical formalism, J. Eur. Math. Soc., Volume 20 (2018) no. 10, pp. 2357-2412 | DOI | MR | Zbl

[Hof77] Hofbauer, Franz Examples for the nonuniqueness of the equilibrium state, Trans. Am. Math. Soc., Volume 228 (1977), pp. 223-241 | DOI | MR | Zbl

[Jen06] Jenkinson, Oliver Every ergodic measure is uniquely maximizing, Discrete Contin. Dyn. Syst., Volume 16 (2006) no. 2, pp. 383-392 | DOI | MR | Zbl

[KH95] Katok, Anatole; Hasselblatt, Boris Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and Its Applications, 54, Cambridge University Press, 1995 | DOI | Zbl

[KP02] Krantz, Steven G.; Parks, Harold R. A primer on real analytic functions, Birkhäuser Advanced Texts. Basler Lehrbücher, Birkhäuser, 2002 | DOI | Zbl

[KW15] Kucherenko, Tamara; Wolf, Christian Localized pressure and equilibrium states, J. Stat. Phys., Volume 160 (2015) no. 3, pp. 1529-1544 | DOI | MR | Zbl

[Lep15] Leplaideur, Renaud Chaos: butterflies also generate phase transitions, J. Stat. Phys., Volume 161 (2015) no. 1, pp. 151-170 | DOI | MR | Zbl

[LW19] Leplaideur, Renaud; Watbled, Frédérique Generalized Curie–Weiss model and quadratic pressure in Ergodic Theory, Bull. Soc. Math. Fr., Volume 147 (2019) no. 2, pp. 197-219 | DOI | MR | Zbl

[LW20] Leplaideur, Renaud; Watbled, Frédérique Curie–Weiss type models for general spin spaces and quadratic pressure in ergodic theory, J. Stat. Phys., Volume 181 (2020) no. 1, pp. 263-292 | DOI | MR | Zbl

[Mis77] Misiurewicz, Michal A short proof of the variational principle for a + N action on a compact space, International Conference on Dynamical Systems in Mathematical Physics (Rennes, 1975) (Astérisque), Volume 40, Société Mathématique de France, 1977, pp. 147-157 | Zbl

[Mit15] Mityagin, Boris S. The zero set of a real analytic function (2015) (

[MN05] Melbourne, Ian; Nicol, Matthew Almost sure invariance principle for nonuniformly hyperbolic systems, Commun. Math. Phys., Volume 260 (2005) no. 1, pp. 131-146 | DOI | MR | Zbl

[Nic11] Nicolaescu, Liviu I. An Invitation to Morse Theory, Universitext, Springer, 2011 | DOI | Zbl

[Oli98] Olivier, Eric Analyse multifractale de fonctions continues, C. R. Math. Acad. Sci. Paris, Volume 326 (1998) no. 10, pp. 1171-1174 | DOI | MR | Zbl

[Ols03] Olsen, Lars Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages, J. Math. Pures Appl., Volume 82 (2003) no. 12, pp. 1591-1649 | DOI | MR | Zbl

[Phe01] Phelps, Robert R. Lectures on Choquet’s Theorem, Lecture Notes in Mathematics, 1757, Springer, 2001 | DOI | MR | Zbl

[PS07] Pfister, Charles-Edouard; Sullivan, Wayne G. On the topological entropy of saturated sets, Ergodic Theory Dyn. Syst., Volume 27 (2007) no. 3, pp. 929-956 | DOI | MR | Zbl

[Rei65] Reif, F. Fundamentals of Statistical and Thermal Physics, McGraw-Hill, 1965

[Roc70] Rockafellar, R. Tyrell Convex analysis, Princeton Mathematical Series, 28, Princeton University Press, 1970 | DOI | Zbl

[Rue04] Ruelle, David Thermodynamic formalism. The mathematical structures of equilibrium statistical mechanics, Cambridge Mathematical Library, Cambridge University Press, 2004 | Zbl

[Rén57] Rényi, Alfréd Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hung., Volume 8 (1957), pp. 477-493 | DOI | MR | Zbl

[Sar01] Sarig, Omri M. Phase Transitions for Countable Topological Markov Shifts, Commun. Math. Phys., Volume 217 (2001) no. 3, pp. 555-577 | DOI | MR | Zbl

[Sar15] Sarig, Omri M. Thermodynamic formalism for countable Markov shifts, Hyperbolic dynamics, fluctuations and large deviations. Special semester on hyperbolic dynamics, large deviations and fluctuations (Proceedings of Symposia in Pure Mathematics), Volume 89, American Mathematical Society, 2015, pp. 81-117 | MR | Zbl

[Sin72] Sinai, Ya Gibbs measures in ergodic theory, Usp. Mat. Nauk, Volume 27 (1972) no. 4 (166), pp. 21-64 | MR | Zbl

[Tha80] Thaler, Maximillian Estimates of the invariant densities of endomorphisms with indifferent fixed points, Isr. J. Math., Volume 37 (1980) no. 4, pp. 303-314 | DOI | MR | Zbl

[TV03] Takens, Flokis; Verbitskiy, Evgeny On the variational principle for the topological entropy of certain non-compact sets, Ergodic Theory Dyn. Syst., Volume 23 (2003) no. 1, pp. 317-348 | MR | Zbl

[Vil09] Villani, Cédric Optimal transport, old and new, Grundlehren der Mathematischen Wissenschaften, 388, Springer, 2009 | DOI | Zbl

[Wal82] Walters, Peter An introduction to ergodic theory, Graduate Texts in Mathematics, 79, Springer, 1982 | Zbl